A differential-geometry approach to operator mixing in massless QCD-like theories and Poincaré-Dulac theorem

Abstract

We review recent progress on operator mixing in the light of the theory of canonical forms for linear systems of differential equations and, in particular, of the Poincaré-Dulac theorem. We show that the matrix $A(g)=-\frac{\gamma (g)}{\beta (g)}=\frac{{\gamma }_0}{{\beta }_0}\frac{1}{g}+\cdots$ determines which different cases of operator mixing can occur, and we review their classification. We derive a sufficient condition for $A(g)$ to be set in the one-loop exact form $A(g)=\frac{{\gamma }_0}{{\beta }_0}\frac{1}{g}$. Finally, we discuss the consequences of the unitarity requirement in massless QCD-like theories, and we demonstrate that ${\gamma }_0$ is always diagonalizable if the theory is conformal invariant and unitary in its free limit at $g=0$.

8 pages, fixed typo in the references. Contribution to the 15th International Symposium on Radiative Corrections (RADCOR) and the XIX Workshop on Radiative Corrections for the LHC and Future Colliders (LoopFest). arXiv admin note: substantial text overlap with arXiv:2103.16220